Optimality and Risk - Modern Trends in Mathematical Finance (eBook)

Preface 7
Contents 11
Contributors 16
On the Extension of the Namioka-Klee Theorem and on the Fatou Property for Risk Measures 18
Introduction 18
The Extended Namioka Theorem 24
The Current Literature 26
On Order Lower Semicontinuity in Riesz Spaces 27
Equivalent Formulations of Order l.s.c. 28
The Order Continuous Dual Xn 29
On the C-Property 30
The C-Property in the Representation of Convex and Monotone Functionals 31
Orlicz Spaces and Applications to Risk Measures 33
Orlicz Spaces Have the C-Property 33
New Insights on the Downside Risk and Risk Measures Associated to a Utility Function u 36
Quadratic-Flat Utility 43
Exponential Utility 44
References 45
On Certain Distributions Associated with the Range of Martingales 46
Introduction 46
Proofs 50
Conclusion 54
References 55
Differentiability Properties of Utility Functions 56
Notation and Preliminaries 56
The Jouini-Schachermayer-Touzi Theorem 59
A Consequence of Ekeland's Variational Principle and Other Family Members of Bishop-Phelps 60
A Consequence of Automatic Continuity 61
The One-Sided Derivative 61
An Example 62
The Example of an Incomplete Financial Market 65
References 65
Exponential Utility Indifference Valuation in a General Semimartingale Model 66
Introduction 66
Motivation and Definition of FER(H) 68
No-arbitrage and existence of FER(H) 73
Relating FER(H) and FER(0) to the Indifference Value 81
A BSDE Characterization of the Indifference Value Process 90
Application to a Brownian Setting 99
References 102
The Expected Number of Intersections of a Four Valued Bounded Martingale with any Level May be Infinite 104
Introduction 104
Proof of Theorem 2. Cases N=2 and N=3 107
Proof of Theorem 2. Case N> 3. An Example
References 115
Immersion Property and Credit Risk Modelling 116
Introduction 116
Credit Modelling Framework 118
The Two Information Flows 119
Financial Interpretation of This Decomposition 120
Absence of Arbitrage 121
Representation Theorem in the Enlarged Filtration 124
Representation of the G-Martingales 124
Change of Probability 129
Complete Reference Market 130
Description of the G-Martingale Probabilities 130
Completeness of the Full Market 132
The Default-Sensitive Asset 133
The Unique Martingale Probability 133
Immersion Property 135
Immersion and Completeness in an Arbitrage Free Set Up 135
Case Where the Credit Event is an Initial Time 136
Credit Risk Premium 138
Incomplete Markets 138
The Risk-Neutral Probabilities of the Full Market 139
Default-Free Pricing Invariance 141
Immersion Property 146
Conclusion 146
References 146
Optimal Consumption and Investment with Bounded Downside Risk for Power Utility Functions 150
Introduction 150
Formulating the Problem 152
The Model 152
The Control Processes 154
The Cost Functions 155
The Downside Risk Measures 156
Problems and Solutions 158
The Unconstrained Problem 158
Value-at-Risk as Risk Measure 160
Expected Shortfall as Risk Measure 165
Proofs 167
Proof of Theorem 1 167
Proof of Theorem 2 167
Proof of Theorem 3 170
Proof of Theorem 4 172
Proof of Theorem 5 172
Proof of Lemma 1 175
Proof of Theorem 7 175
Proof of Theorem 8 176
Proof of Theorem 9 176
Appendix 178
A Technical Lemma 178
The Verification Theorem 178
A Special Version of Itô's Formula 182
References 186
On Comparison Theorem and its Applications to Finance 188
Introduction 188
Comparison Theorem 189
Applications to Mathematical Finance 194
References 197
Examples of FCLT in Random Environment 199
Introduction 199
1. 199
2. 200
3. 202
4. 202
Assumptions, Notations and Main Result 202
Notations 202
Assumptions 202
The Proof of Theorem 1 203
Auxiliary Lemma 203
The Proof of (2) 204
The Tightness {supt< =T|Xepsilont|}epsilon->
The Proof of (13)-(15) 206
Diffusion in Random Environment 207
b(omega,u)0 207
b(omega,u)0 208
Markov Chain as Random Environment 208
Langevin Random Environment 209
References 210
The Optimal Time to Exchange one Asset for Another on Finite Interval 213
Introduction 213
Basic Properties of Premium Function and Stopping Domain 214
Integral Equations for the Premium Function and the Threshold Curve 217
Approaching Solution of Integral Equation for Threshold Curve 220
References 226
Arbitrage Under Transaction Costs Revisited 227
Introduction 227
Arbitrage and Price Systems 228
Markets with One Risky Asset 231
Proofs 233
Conclusion 236
Appendix 236
References 240
On the Linear and Nonlinear Generalized Bayesian Disorder Problem (Discrete Time Case) 242
Linear Penalty Case 242
1. 242
2. 243
3. 244
Nonlinear Penalty Case 245
1. 245
2. 245
3. 247
4. 248
5. 249
References 250
Long Time Growth Optimal Portfolio with Transaction Costs 252
Introduction 252
Discrete Time Case 256
Continuous Time Case 260
References 264
On the Approximation of Geometric Fractional Brownian Motion 266
Introduction 266
Geometric Fractional Brownian Motion 266
Motivation 267
The Structure of the Note 268
Approximation of fBm 268
Construction of the Approximation 268
Further Properties of the Approximation 269
Approximation to Geometric fBm 272
Some Properties of the Approximation 272
Set-Up 272
Prelimit Market Models are Arbitrage-Free 273
Prelimit Market Models are Complete 275
Discussion and Conclusion 278
References 280